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A probability space is a mathematical triplet ... However, the probability of the union of an uncountable set of events is not the sum of their probabilities.
However, in this case it is no longer true that a finite Baire measure is necessarily regular: for example, the Baire probability measure that assigns measure 0 to every countable subset of an uncountable discrete space and measure 1 to every co-countable subset is a Baire probability measure that is not regular.
The best known example of an uncountable set is the set of all real numbers; Cantor's diagonal argument shows that this set is uncountable. The diagonalization proof technique can also be used to show that several other sets are uncountable, such as the set of all infinite sequences of natural numbers (see: (sequence A102288 in the OEIS)), and the set of all subsets of the set ...
It can also be shown that the Haar measure is an image of any probability, making the Cantor set a universal probability space in some ways. In Lebesgue measure theory, the Cantor set is an example of a set which is uncountable and has zero measure. [16] In contrast, the set has a Hausdorff measure of 1 in its dimension of log 2 / log 3. [17]
In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra"; also σ-field, where the σ comes from the German "Summe" [1]) on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections. The ordered pair (,) is called a measurable space.
A probability space is a measure space with a probability measure. For measure spaces that are also topological spaces various compatibility conditions can be placed for the measure and the topology. Most measures met in practice in analysis (and in many cases also in probability theory) are Radon measures.
An important example, especially in the theory of probability, is the Borel algebra on the set of real numbers.It is the algebra on which the Borel measure is defined. . Given a real random variable defined on a probability space, its probability distribution is by definition also a measure on the Borel a
Probability is the branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur. [note 1] [1] [2] This number is often expressed as a percentage (%), ranging from 0% to ...